The Horseshoe prior is a widely acclaimed choice as a shrinkage prior owing to its infinite spike at 0 and simultaneously possessing a thick tail. The tails decay like a second-order polynomial, and hence the corresponding penalty function behaves like a logarithmic penalty, and is non-convex (see [1]}, [2]} for more details). The Horseshoe prior can be expressed as a scale mixture of normals with half-Cauchy prior, thus acting as a global-local shrinkage prior. We put a Horseshoe prior on the pairwise differences in the parameters \(\eta _i = \theta _i - \theta _{i-1},\; i = 2,\ldots ,n.\)  We also need to put suitable priors on the mean parameter \(\theta _1\)  and the error variance \(\sigma ^2\) . The full prior specification is given by,
\(\theta _1 \mid \lambda _1^2, \sigma ^2 \sim N(0, \lambda _1^2\sigma ^2),\; \eta _i \mid \lambda _i^2,\tau ^2, \sigma ^2 \stackrel{ind}{\sim } N(0, \lambda _i^2\tau ^2\sigma ^2), \; 2 \le i \le n,\nonumber \\\lambda _i \stackrel{iid}{\sim } C^+(0,1),\; 2 \le i \le n,\; \tau \sim C^+(0,1),\; \sigma ^2 \sim IG(a_\sigma , b_\sigma ).\) 
